Influence of protein patchiness and salinity on complexation

We will now investigate the effect of changing the charge distribution on the surface of the globular protein as well as the ionic strength on the effective interaction in more detail.

Results for one-patched proteins

The results of protein-PE simulations with one patch, P_s¹, and growing patch charge s = 8,12,16 at 20 mM salt concentration are shown in Figure 5.18, respectively. The PMFs are now strongly attractive in the tens of kBT for a wide distance-range except for a small repulsive barrier at aroundr≈6nm. For rising patch charge s (and thus increasing dipole) of the patchy protein the attraction is more pronounced and the barrier vanishes. The insets in Table 5.2 show typical configurations of the protein-PE complex in the stable bound state at r ≈ 2.5 nm. Here, we can see that almost the entire PE chain is adsorbed on the patch in the bound state. Configurations for other distances are also displayed in Table 5.2. For r ≈ 10 nm the PE is desorbed and exhibits relatively stiff, rod-like configurations. For a closer distance of about r ≈ 6 nm the PE is able to reach out and touch the attractive patch; this is reflected in the onset of attraction in the PMFs in Figure 5.18 (a). The distance r ≈ 1.5 nm in Table 5.2 corresponds to the closest distance approachable in our SLD simulations and is, according to the PMF, energetically strongly penalized. Here, the PE embraces the globular protein to fulfill the external force constraint in the SLD that the PE center of mass is close to that of the globular protein.

The number of negatively charged ions (NCI) as well as positively charged ions (PCI) con-densed on the protein patch and on the PE for the [P_s¹−P E₂₅] simulations are shown in Figure 5.18 (b). At large separations the number of condensed ions is fairly constant, while the absolute value increases with growing patch charges. When the PE begins to adsorb on the patch, r6 nm, counterions and coions on both molecules are simultaneously released, their number increasing the closer the associating partners come to each other. Hence, protein-PE complexation is accompanied by a significant release of condensed ions mostly stemming from the PE. A detailed analysis on numbers will follow in the next paragraph.

In Figure 5.18 (c) the patch orientations along the separation r with respect to the distance vector are presented. For large distances (r > 7 nm) no correlation effect is observable, however, when the PE begins to attach to the patch with its first monomer, a favorite orien-tation of the patch towards the PE immediately locks in. This orienorien-tation persists until the PE is completely attached to the patch. For very small distances r 2 nm the orientation correlation weakens due to the (forced) embracing of the PE around the globular protein.

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Figure 5.19: Simulation of [P₁₂¹ −P E₂₅] at different salt concentrations ranging from 10 mM to 200 mM.

(a), (b), and (c) are the same as in Figure 5.18.

Effects of added salt

The influence of the ionic strength on the PMF and the ion release is investigated based on the [P₁₂¹ − P E₂₅] complex. The results for salt concentrations between 10 mM and 200 mM are presented in Figure 5.19. First of all, it is evident that the attraction of the PMFs as depicted in Figure 5.19 (a) decreases with increasing salt concentration c_s since the electrostatic interaction between the patch and the PE monomers is more screened. A further effect of the screening is that the beginning of the attraction (adsorption of the PE head monomer to the patch) is shifted to shorter separations with increasing c_s due to a lesser stiffness of the PE chain. The corresponding number of condensed ions N_c is shown in Figure 5.19 (b), respectively. It is clear that increasing the ionic strength leads to more condensed ions on both molecules which can be unambiguously verified from the trajectories.

The average number of released ions is between 4.4 for the lowest salt concentration and up to 7.8 for the highest salt concentration. The position of the N_c-minimum roughly coincides with the PMF minimum, i.e., there is a clear correlation between ion release and attraction for all salt concentrations. The patch orientation behavior, cf. Figure 5.19 (c), does not exhibit any marked salt concentration behavior, apart from a shorter correlation range for increasing c_s due to the shorter attraction range discussed above.

-35 -30 -25 -20 -15

200

10 100

βwmin(cs)

c_s [mM]

Simulation CR/DH-model 4.9⋅ln[c_s]

Figure 5.20: The binding affinity represented by the minimum value of the PMF,w_min, of the[P₁₂¹ −P E₂₅] complex as a function of the salt concentrationc_s (violet symbols) in a lin-log plot. The linear blue solid line is a fit according to the fit functionβw_min(c_s) = ˜a+ ˜Nln[c_s]with ˜a=−44.3 andN˜ = 4.9. The green solid line is a fit to the combined CR/DH model around Eq. (5.5), see text for explanation.

Salt concentration of the binding affinity

To analyze the correlation between the free energy of bindingw_min(the ’binding affinity’) and salt concentration in more detail, we plot in Figure 5.20 the variation of the PMF minimum with the logarithm of the salt concentration ln[c_s]. As motivated by our discussion on ion release effects around Eq. (3.16) in Section 3.1.4, we have fitted the data with a function of the form βw_min(c_s) = ˜a+ ˜Nln[c_s]. The result is also shown in Figure 5.20 and represents actually a very satisfactory fit to the data with ˜a=−44.3andN˜ = 4.9, implying that every time an average of 4.9 ions are released upon complexation, independent of salt.

We go further into the details of the counterion release analysis by actually counting ions and evaluating the free energy changes corresponding to ions released from patch and PE as defined in Eq. (3.17), respectively. The results are summarized in Table 5.3. First we would like to direct the attention to the value of N₊^apart, the number of condensed ions on the PE chain in the isolated state. With a Manning parameter of Γ = 1.78 we expect N_mon(1−Γ⁻¹) = 10.9counterions to be condensed right at the chain. As we see in Table 5.3, we indeed find numbers between 9.7 at the lowest salt concentration up to 13.6 at the highest c_s, coinciding with the prediction but also exhibiting a noticeable salt dependence.

The number of released ions from the chain upon complexation, ΔN₊ only slightly depends on salt, increasing from roughly 3.5 to 4.7. This also agrees well with a prediction that Z_s·N_mon = 5.1, with the binding patch valence Z_s = 12, should be released from the PE upon binding. Hence, the ion-PE system behaves as expected and shows a very robust ion condensation and release effect.

In contrast to the ionic behavior at the PE, the number of accumulated and released ions on the protein patch, ΔN₋, increases from about 1 to 3.2 in the considered concentration range, i.e., a much stronger salt dependence of the number of released ions is found at the patch. The individual free energy contributions w_patch and w_PE, evaluated by Eq. (3.17),

Table 5.3: A summary of the values of the PMF minimum w_min with respect to the salt concentration c_s and the number of condensed ions when the patchy protein and PE are apart (N_i^apart) and in the complexed state (N_i^min) taken from the data for[P₁₂¹ −P E₂₅]in Figure 5.19 and 5.18. The difference of condensed ions is then ΔN_i =N_i^apart−N_i^min. The concentrationc_patchis the local density of negative ions on the protein patch. Forc_{P E} in Eq. (3.17) we used a constant 3.5 M as measured in our simulation.

c_s w_min Patch PE

[mM] [kBT] N₋^apart N₋^min ΔN₋ c_patch [M]βw_patch N₊^apart N₊^min ΔN₊ βw_PE

10 -32.3 0.96 0.00 0.96 1.36 -4.7 9.69 6.24 3.45 -20.2

20 -30.3 1.41 0.00 1.41 1.99 -6.5 10.39 6.62 3.77 -19.4

50 -25.3 2.09 0.02 2.07 2.96 -8.5 11.35 7.22 4.13 -17.5

100 -21.9 2.65 0.05 2.60 3.75 -9.4 12.42 7.93 4.49 -16.0 200 -17.6 3.31 0.15 3.16 4.68 -10.0 13.59 8.92 4.67 -13.4 are also shown in Table 5.3. Attractive contributions from the patch are actually growing (from about −5to−10kBT) for increasing c_s due to the significant increase of accumulated ions on the patch. In contrast, the contribution from the PE is decreasing (from about −20 to −13kBT) because the number of condensed ions stays relatively constant. Interestingly, the sum of both contributions only shows little salt dependence and is about −25± 1 kBT, while the simulated free energy is about 14 kBT from ca. 32 kBT at 10 mM to ca. 18 kBT at 200 mM. Clearly, the approach Eq. (3.17) cannot satisfactorily describe the values and trends of the binding affinity with salt concentration. We believe that this must be assigned to a missing counterion condensation mechanism on the protein patch, where only conventional charge screening effects apparently play a role. We note that we have experimented with the cut-off radii that define the condensed layer around the patch in a reasonable range but have not found any qualitative improvement of the prediction.

The charge screening effect should be captured in our combined CR/DH model, Eq. (5.5), as previously introduced. A best fit is also presented in Figure 5.20. Since we find on average about 8 ions still condensed on the complexed PE, the PE charge Q_{P E} was fixed by an effective valence of −25−(−8) = −17. The number of condensed ions N˜ on the PE was fixed to 4.9 as found in the Record-Lohman fit. The only remaining fit parameter is the effective size of the adsorbed chain, R_{P E}. For the best fit we find R_{P E} = 1.17 nm. This value is indeed close to the patch size and to the mean radius of gyration of about 1.3 nm in the bound state (cf. Figure 5.21). As we can see in Figure 5.20 the fit describes the simulated data very well with essentially only one fit parameter of reasonable value. It is interesting to see that the DH part of the theory induces some curvature to the fitted curve in addition to the linear logarithmic behavior in this lin-log plot. Actually such a curvature can also be noticed made by the simulation data points. Hence, we have strong indications that the combined CR/DH model captures the right physics in the system, in particular the fact that the condensed ions on the PE onlyplay the decisive role in the counterion release framework.

In a recent experimental and computational study of Yu et al. [Paper V], it was found that counterion release occur even in more complex protein models and essentially contribute to the binding affinity. This also justifies the use of our patchy protein models.

2.0

1.2 1.4 1.6 1.8 2.2

1 2 3 4 5 6 7 8 9 10 11

RPE [nm]

r [nm]

Figure 5.21: The mean radius of gyration R_{P E} of the 25meric PE chain resolved versus the center of mass distance to theP₁₂¹ protein at a salt concentration of 50 mM. The PE bulk value of the sizeR_{P E} ≈1.6 nm, increases at intermediate distances (r≈5 nm), where the chain stretches out, to a value ofR_{P E} ≈2.1 nm and collapses toR_{P E} ≈1.3 nm in the bound state (r≈2.4nm).

Results for two-patched proteins

Figure 5.22 shows results of protein-PE complexes with two (m = 2) antipodally aligned patches in an ionic solution of 20 mM. Despite an additional patch, the simulated PMFs between[P_s²−P E₂₅]in Figure 5.22 (a) are less attractive compared to[P_s¹−P E₂₅]systems with s = 8,12 (see Figure 5.18). Furthermore, a distinct shift of the global minimum to a larger separation is present. A possible explanation could be that in patchy protein models with m = 2 the negative charges are denser distributed on the surface leading to a raised repulsive electrostatic interaction and a reduced attraction between PE monomers and the patchy protein. In counting the condensed ionsN_c on the both molecules again, ion release is found as depicted in Figure 5.22 (b) where ions are mainly released from the PE. As in the one-patch systems, the attraction is accompanied by a strong orientation of the interacting attractive patch to the PE, see Figure 5.22 (c). The orientation is reversed at very small distances r 1.8 nm; but note that those are improbable to observe in equilibrium anyway due to the high free energy penalty. Hence, for the short P E₂₅ chain that cannot reach to the second patch of the protein, the results are qualitatively similar than for the analogous one-patch system, albeit with considerably less attraction.